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Register a new userComputing GIT-fans with symmetry and the Mori chamber decomposition of $bar{M}_{0,6}$
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We propose an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry and group theory. We have implemented our algorithm in the Singular library gitfan.lib. Using our implementation, we compute the Mori chamber decomposition of the cone of movable divisors of $bar{M}_{0,6}$.
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The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of smooth tropical varieties. Among the resulting applications, we prove the Hodge isomorphism theorem which asserts that the Chow rings of smooth unimodular tropical fans are isomorphic to the tropical cohomology rings of their corresponding canonical compactifications, and prove a slightly weaker statement for any unimodular fan. We furthermore introduce a notion of shellability for tropical fans and show that shellable tropical fans are smooth and thus enjoy all the nice homological properties of smooth tropical fans. Several other interesting properties for tropical fans are shown to be shellable. Finally, we obtain a generalization, both in the tropical and in the classical setting, of the pioneering work of Feichtner-Yuzvinsky and De Concini-Procesi on the cohomology ring of wonderful compactifications of complements of hyperplane arrangements. The results in this paper form the basis for our subsequent works on Hodge theory for tropical and non-Archimedean varieties.
We present an algorithm to compute the automorphism group of a Mori dream space. As an example calculation, we determine the automorphism groups of singular cubic surfaces with general parameters. The strategy is to study graded automorphisms of affine algebras graded by a finitely generated abelian groups and apply the results to the Cox ring. Besides the application to Mori dream spaces, our results could be used for symmetry based computing, e.g. for Grobner bases or tropical varieties.
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on $h$. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on $h$ than here.
Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set $W$ defined by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant triangular decomposition, we were able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success.
We give a notion of combinatorial proximity among strongly stable ideals in a given polynomial ring with a fixed Hilbert polynomial. We show that this notion guarantees geometric proximity of the corresponding points in the Hilbert scheme. We define a graph whose vertices correspond to strongly stable ideals and whose edges correspond to pairs of adjacent ideals. Every term order induces an orientation of the edges of the graph. This directed graph describes the behavior of the points of the Hilbert scheme under Grobner degenerations with respect to the given term order. Then, we introduce a polyhedral fan that we call Grobner fan of the Hilbert scheme. Each cone of maximal dimension corresponds to a different directed graph induced by a term order. This fan encodes several properties of the Hilbert scheme. We use these tools to present a new proof of the connectedness of the Hilbert scheme. Finally, we improve the technique introduced in the paper Double-generic initial ideal and Hilbert scheme by Bertone, Cioffi and Roggero to give a lower bound on the number of irreducible components of the Hilbert scheme.
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